Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Discover the answers you need from a community of experts ready to help you with their knowledge and experience in various fields. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To determine the first term [tex]\(a\)[/tex] and the common difference [tex]\(d\)[/tex] of an arithmetic series given that the sum of the first 710 terms is 27.5 and the 10th term is 5, we follow a systematic approach:
### Step 1: Define the Formulas
1. The sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms of an arithmetic series can be given by:
[tex]\[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \][/tex]
2. The [tex]\(n\)[/tex]th term of an arithmetic series can be defined as:
[tex]\[ a_n = a + (n-1)d \][/tex]
### Step 2: Substitute the Known Values
Given:
- [tex]\(n = 710\)[/tex]
- [tex]\(S_{710} = 27.5\)[/tex]
- The 10th term [tex]\(a_{10} = 5\)[/tex]
### Step 3: Set Up the Equations
1. Using the given sum of the first 710 terms:
[tex]\[ 27.5 = \frac{710}{2} (2a + 709d) \][/tex]
Simplify this equation:
[tex]\[ 27.5 = 355 (2a + 709d) \][/tex]
[tex]\[ 2a + 709d = \frac{27.5}{355} \][/tex]
2. For the 10th term:
[tex]\[ a + 9d = 5 \][/tex]
### Step 4: Solve the System of Equations
We now have two equations:
1. [tex]\(2a + 709d = \frac{27.5}{355}\)[/tex]
2. [tex]\(a + 9d = 5\)[/tex]
We solve these equations simultaneously.
### Step 5: Find the Values of [tex]\(a\)[/tex] and [tex]\(d\)[/tex]
From the equations, solving for [tex]\(a\)[/tex] and [tex]\(d\)[/tex]:
1. From the second equation:
[tex]\[ a = 5 - 9d \][/tex]
2. Substitute [tex]\(a = 5 - 9d\)[/tex] into the first equation:
[tex]\[ 2(5 - 9d) + 709d = \frac{27.5}{355} \][/tex]
[tex]\[ 10 - 18d + 709d = \frac{27.5}{355} \][/tex]
[tex]\[ 691d + 10 = \frac{27.5}{355} \][/tex]
[tex]\[ 691d = \frac{27.5}{355} - 10 \][/tex]
[tex]\[ d = \frac{\frac{27.5}{355} - 10}{691} \][/tex]
Thus:
[tex]\[ d = -0.0143596746906912 \][/tex]
And substituting [tex]\(d\)[/tex] back into [tex]\(a = 5 - 9d\)[/tex]:
[tex]\[ a = 5 - 9 \times (-0.0143596746906912) \][/tex]
[tex]\[ a = 5.12923707221622 \][/tex]
### Conclusion
The first term [tex]\(a\)[/tex] is approximately [tex]\(5.12923707221622\)[/tex] and the common difference [tex]\(d\)[/tex] is approximately [tex]\(-0.0143596746906912\)[/tex].
### Step 1: Define the Formulas
1. The sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms of an arithmetic series can be given by:
[tex]\[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \][/tex]
2. The [tex]\(n\)[/tex]th term of an arithmetic series can be defined as:
[tex]\[ a_n = a + (n-1)d \][/tex]
### Step 2: Substitute the Known Values
Given:
- [tex]\(n = 710\)[/tex]
- [tex]\(S_{710} = 27.5\)[/tex]
- The 10th term [tex]\(a_{10} = 5\)[/tex]
### Step 3: Set Up the Equations
1. Using the given sum of the first 710 terms:
[tex]\[ 27.5 = \frac{710}{2} (2a + 709d) \][/tex]
Simplify this equation:
[tex]\[ 27.5 = 355 (2a + 709d) \][/tex]
[tex]\[ 2a + 709d = \frac{27.5}{355} \][/tex]
2. For the 10th term:
[tex]\[ a + 9d = 5 \][/tex]
### Step 4: Solve the System of Equations
We now have two equations:
1. [tex]\(2a + 709d = \frac{27.5}{355}\)[/tex]
2. [tex]\(a + 9d = 5\)[/tex]
We solve these equations simultaneously.
### Step 5: Find the Values of [tex]\(a\)[/tex] and [tex]\(d\)[/tex]
From the equations, solving for [tex]\(a\)[/tex] and [tex]\(d\)[/tex]:
1. From the second equation:
[tex]\[ a = 5 - 9d \][/tex]
2. Substitute [tex]\(a = 5 - 9d\)[/tex] into the first equation:
[tex]\[ 2(5 - 9d) + 709d = \frac{27.5}{355} \][/tex]
[tex]\[ 10 - 18d + 709d = \frac{27.5}{355} \][/tex]
[tex]\[ 691d + 10 = \frac{27.5}{355} \][/tex]
[tex]\[ 691d = \frac{27.5}{355} - 10 \][/tex]
[tex]\[ d = \frac{\frac{27.5}{355} - 10}{691} \][/tex]
Thus:
[tex]\[ d = -0.0143596746906912 \][/tex]
And substituting [tex]\(d\)[/tex] back into [tex]\(a = 5 - 9d\)[/tex]:
[tex]\[ a = 5 - 9 \times (-0.0143596746906912) \][/tex]
[tex]\[ a = 5.12923707221622 \][/tex]
### Conclusion
The first term [tex]\(a\)[/tex] is approximately [tex]\(5.12923707221622\)[/tex] and the common difference [tex]\(d\)[/tex] is approximately [tex]\(-0.0143596746906912\)[/tex].
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
